ACOUSTIC IMPEDANCE. 29 



Appendix. 



On ihr Aiialoyics between Sound-]]'ave Tran.smi.s-sion along a Utiiform 

 Acoustic Tube Conductor, and Electromagnet ic-]V are Transmission 

 along a Uniform Electric Line Conductor. 



We may assume that when a telephone receiver, actuated as an 

 acoustic generator, sings a pure tone steacUly into a long smooth 

 straight tube of S sq. cm. internal cross section, arranged as in Figure 

 2, the sound waves tra\'elling along the tube are plane waves, and that 

 the effects on them of friction at the walls of the tube may be neglected. 

 E\en if the actual conditions do not conform closely with these 

 postulates, the results thus arrived at may safely be regarded as first 

 approximations. 



The distance of any cross-section of the tube, measured along the 

 axis, from the generator diaphragm may be expressed as 1 cm. The 

 clamping-circle diameter of the receiver diaphragm may first be sup- 

 posed to be equal to the internal diameter of the tube. The very 

 small \ibratory displacement of the air, in a sound wave, from its 

 normal quiescent position, measured parallel to the axis, in the direc- 

 tion of increasing 1, may be denoted by x cm. Since x, in the assumed 

 case of a pure tone and simple harmonic vibration, may be taken as 

 the projection, at time t seconds, on the tube axis, of the rotating 

 plane vector 



X = .r„i e-'"" instantaneous cm.Z (22) 



where w is the impressed angular Aelocity of rotation, and .r,„ is the 

 maximum cyclic vibratory displacement, the vibratory velocity of a 

 transverse layer of air in the tube, also measured along the axis, and 

 in the direction of increasing 1, is 



^•'' • ■ oj/ • ;,, inst. kines or ,^.^> 



3t =•'•= .Z-^"^^^' = .'-■=. .•„.-' ^^j^^^^ (23) 



The velocity x is also a planevector ([uantity, capable of being dealt 

 with by the rules of complex arithmetic. Likewise, the vibratory 

 acceleration will be 



^^^ ■■ • • i^t ■ ■ - i^t - inst. kines/sec. , 



— = .r = ju.x„,e^ = jcx = - co-.r,„e^ '= - co-.r ^^ ^^j^,^.^ (24) 



